Question

Indicate whether each statement is true or false: $$ \begin{array}{l}{\text { (a) The band gap of a semiconductor decreases as the parti- }} \\ {\text { cle size decreases in the } 1-10 \text { -nm range. }} \\\ {\text { (b) The light that is emitted from a semiconductor, upon }} \\\ {\text { external stimulation, becomes longer in wavelength as }} \\ {\text { the particle size of the semiconductor decreases. }}\end{array} $$

Short answer

(a) FALSE: The band gap of a semiconductor increases as the particle size decreases, due to the quantum confinement effect. (b) FALSE: As the particle size of the semiconductor decreases, the band gap increases, leading to higher energy emitted light. Consequently, the wavelength of the emitted light becomes shorter, not longer, as the particle size decreases.

Step-by-step solution

Band Gap and Semiconductor

A semiconductor is a material with a band gap that separates the valence band (occupied by electrons) from the conduction band (where there are no electrons, due to the energy gap). The band gap refers to the energy difference between the top of the valence band and the bottom of the conduction band in the material. The smaller the particle size of the semiconductor, the larger the band gap due to quantum confinement effect. Given this relation, (a) The band gap of a semiconductor decreases as the particle size decreases in the 1-10 - nm range.

Answer for (a)

The statement (a) is FALSE as the band gap of a semiconductor increases as the particle size decreases. Understanding the relation between particle size and the wavelength of light emitted upon external stimulation in semiconductors:

Wavelength and Semiconductor

When a semiconductor is externally stimulated, electrons can be excited from the valence band to the conduction band. When the excited electrons relax back to the valence band, energy is released in the form of light. The energy of the light emitted will be equal to the band gap of the semiconductor. As we know that the energy of light is inversely proportional to the wavelength, i.e., \(E = \frac{hc}{λ}\), where h is Planck's constant, c is the speed of light, and λ is the wavelength of light. Given this relation, (b) The light that is emitted from a semiconductor, upon external stimulation, becomes longer in wavelength as the particle size of the semiconductor decreases.

Answer for (b)

The statement (b) is FALSE because as the particle size of the semiconductor decreases, the band gap increases, which means the energy of the emitted light increases. Since the energy of light is inversely proportional to the wavelength, the wavelength of the emitted light will be shorter, not longer, as the particle size decreases. Final answers: (a) FALSE (b) FALSE

Key concepts

Quantum Confinement Effect

In the fascinating world of nanotechnology, the quantum confinement effect plays a crucial role in tailoring the properties of semiconductors. This phenomenon occurs when the size of a semiconductor particle is reduced down to a nanometer scale (typically less than 10 nm), which is comparable to the de Broglie wavelength of electrons.

Imagine confining an electron within a space so small that it's comparable to the electron's own wavelength. This restriction fundamentally alters the electron's energy states and, as a result, increases the semiconductor's band gap. The band gap is the energy required to elevate an electron from the valence band, where electrons are bound to atoms, to the conduction band, where they are free to move and conduct electricity.

Due to quantum confinement, as the particle size decreases, the available energy levels for the electron become more discrete and widely spaced. This leads to a larger band gap, which directly impacts the optical and electrical properties of the material. For semiconductors, this can result in more efficient and tunable electronic devices, as the wavelength of the light they emit can be manipulated by simply changing the size of the particles.

Electron Excitation in Semiconductors

When discussing semiconductors, electron excitation is an essential concept that explains how these materials respond to external energy inputs, such as light or electric fields. Electrons in a semiconductor are initially located in an energy band known as the valence band. With sufficient energy, these electrons can transition to the conduction band, leaving behind 'holes' in the valence band.

This leap from the valence band to the conduction band is called electron excitation, and it is fundamental to how semiconductors are used in electronics. For instance, in photovoltaic cells, electrons are excited by photons coming from sunlight. In light-emitting diodes (LEDs), when the excited electrons return to the valence band, they release their excess energy in the form of light—causing the LED to emit light.

The essential factor to consider is the band gap: only photons with energy equal to or greater than the band gap can excite electrons. This principle ensures that each semiconductor responds to a specific range of light wavelengths, making semiconductor devices highly adjustable for various applications.

Wavelength and Energy Relationship

The relationship between the wavelength of light and its energy is fundamental in understanding semiconductors' behavior upon stimulation. According to the Planck-Einstein relation, the energy (\(E\)) of a photon is directly proportional to its frequency (\(ν\)) and inversely proportional to its wavelength (\(λ\)): \(E = hν = \frac{hc}{λ}\), where \(h\) is Planck's constant and \(c\) is the speed of light in a vacuum.

This inverse relationship means that as the wavelength of light increases, its energy decreases, and vice versa. In the context of semiconductors, when electrons fall from the high-energy conduction band to the lower-energy valence band, they emit photons. The wavelength of this emitted light directly correlates to the semiconductor's band gap—the larger the band gap, the higher the photon's energy, and the shorter its wavelength. Conversely, a smaller band gap would emit light with a longer wavelength and lower energy.

Therefore, by adjusting the semiconductor's band gap through means such as the quantum confinement effect, one can control the wavelength and energy of the light that semiconductors can emit. This interplay is the cornerstone of designing optical devices like LEDs, lasers, and photodetectors with specific color (wavelength) requirements.